line MN is perpendicular to op
If the coordinates of M,N and O are (2,4), (-5,-8) , (-6,2) respectively, find the equation for line OP
slope of MN = (-8-4)/(-5-2) = 12/7
so slope of perpendicular = - 7/12
equation of perpendicular through (-6,2) is
y-2 = (-7/12)(x+6)
etc
The slope on line MN is (-8-4)/(-5-2) = 12/7. The slope of OP is therefore -7/12.
Find the line in y = mx = b form that has m = 7/12 and passes through O (-6,2)
2 = [(7/12)*(-6] + b
b = 2 + 3/2 = 7/2
y = (7/12)x + 7/2
Check my work
To find the equation for line OP, we need to first find the slope of line OP, and then use one of the given points (O or P) to find the y-intercept.
Since line MN is perpendicular to line OP, we can use the fact that perpendicular lines have slopes that are negative reciprocals of each other.
Step 1: Find the slope of line MN
The slope of line MN can be calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of M and N respectively.
Using the coordinates of M (2,4) and N (-5,-8), we have:
slope of MN = (-8 - 4) / (-5 - 2) = -12 / -7 = 12/7
Step 2: Find the slope of line OP
Since line OP is perpendicular to line MN, the slope of line OP will be the negative reciprocal of the slope of MN. So,
slope of OP = -1 / (12/7) = -7/12
Step 3: Use a point on line OP to calculate the y-intercept
We can use either point O (-6,2) or point P (unknown) to find the y-intercept. Let's choose point O (-6,2).
We have the slope of line OP (-7/12) and the coordinates of point O (-6,2).
Using the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) are the coordinates of the point.
Substituting the values, we get:
y - 2 = (-7/12)(x - (-6))
Simplifying the equation:
y - 2 = (-7/12)(x + 6)
y - 2 = (-7/12)x - 7/2
y = (-7/12)x - 7/2 + 2
y = (-7/12)x - 7/2 + 4/2
y = (-7/12)x - 3/2
Therefore, the equation for line OP is y = (-7/12)x - 3/2.